1. Field
This invention relates to borehole geophysics and inverse geophysical modeling techniques. It provides a method for adapting borehole geophysical techniques to accumulate a mass of redundant resistivity data for inverse mathematical processing into either two-dimensional or three-dimensional models of subsurface resistivity structures.
2. State of the Art
Electrical resistivity surveys are used routinely in geothermal, mineral, coal, groundwater and engineering applications. They are also used in connection with oil and gas exploration. They have recently been used for sensing buried wastes and waste migration.
Resistivity surveys are capable of mapping overburden, depth, dip, depth extent, strike, stratigraphy, faults, fractures, ore deposits, rock units, saltwater intrusion, contaminant plumes, landfills, voids and other subsurface features.
The induced polarization method (which involves applying a potential difference between spaced electrodes) was developed for detecting small concentrations of disseminated mineralization (a target) in base metal exploration. This method has also found limited use for detecting other exploration targets, e.g., in groundwater exploration, geotechnical and environmental applications.
Surface methods (those which rely upon the placement of apparatus at various locations on the surface of the earth) typically suffer from limited depth of penetration and spatial resolution, particularly when the target is deeply buried. Techniques which utilize a borehole within or near an exploration target offer increased depth of penetration and improved vertical and lateral resolution, provided borehole conditions, physical property contrasts, and geological conditions are favorable. Among other reasons, the economic advantages of increased vertical resolution and target discrimination have provided an impetus for improving borehole instrumentation, survey techniques, and the numerical computation of various target responses.
The economic subsurface development of geothermal systems, mineral deposits, oil and gas fields, and groundwater is critically dependent upon a cost-effective drilling program and the successful delineation of a target zone (e.g., fracture zones) within a large region. With existing electrical geophysical methods, satisfactory delineation of such subterranean features is not easily achieved. Reliable delineation of subsurface features associated with landfills, saltwater intrusion, chemical contaminant plumes and voids is of major environmental concern.
The resistivity of a conductor, determined by measuring the potential between two electrodes when a known current is impressed through the conductor, may be expressed by the equation: EQU .rho.=K.multidot.V/I (1) where:
V represents the potential drop in volts across the conductor; PA1 I represents the current flow in amperes through the conductor; and PA1 K is a constant reflecting the geometry of the conductor and the electrode positions. PA1 r.sub.l, r.sub.2, r.sub.3, and r.sub.4 are measured distances between the respective current and potential electrodes according to a prescribed convention, wherein r.sub.l is the distance between a positive current electrode and a first potential electrode, r.sub.2 is the distance between that positive current electrode and a second potential electrode, r.sub.3 is the distance between a negative current electrode and the first potential electrode, and r.sub.4 is the distance between that negative current electrode and the second potential electrode. PA1 "The ingenious method of least squares makes it possible to adjust an arbitrarily overdetermined and incompatible set of equations. In fact, we make an asset out of a liability and try to overdetermine a set of equations as much a possible by making an arbitrary number of surplus observations beyond the minimum number demanded by the number of unknowns.
Equation (1) is readily applied to simple conductors of homogeneous properties, but becomes very complicated in the case of non-homogeneous regions of the earth. It is well recognized that the impressed potential which can be measured in a geophysical survey reflects the apparent resistivity of the earth along the path of the current impressed between and about two spaced current electrodes. An apparent resistivity value can be calculated for any two potential electrodes when the potential drop (V) is measured for a known impressed current (I) between two current electrodes, provided the relative positioning of the electrodes is known. The simple-case calculation assumes a homogeneous earth. Given that assumption, apparent resistivity is determined by the following equation: ##EQU1## .rho.a is the apparent resistivity of the earth between two potential electrodes; and
A distinction exists between electrical well-logging and electrical borehole geophysical survey techniques. Electrical well-logging typically involves the use of several electrodes, spaced a fraction of a meter to several meters apart, located on a sonde. The sonde is lowered into a single well or borehole, and exploration is made of an annulus surrounding the well. The annulus has a radius ranging from a fraction of a meter to tens of meters. In borehole geophysical survey procedures, electrodes of widely varying separation are disposed variously in one or more boreholes and also at the surface of the earth. The exploration range from a single borehole may exceed 100 meters, while exploration between boreholes may exceed several hundreds of meters.
There are several known modes of electrical borehole geophysical surveying. By the convention followed in this disclosure, the location of the energizing (current) electrode is first designated, followed by the location of the measurement (potential) electrodes. Thus, in the surface-to-borehole method, the energizing electrodes are disposed at horizontal locations on the surface of the earth with only potential electrodes lowered to occupy numerous vertical locations down one or more boreholes. In the borehole-to-surface method, the energizing electrodes occupy one or more vertical locations in a borehole while the potential electrodes occupy numerous horizontal locations on the surface of the earth. In the cross-borehole method, the energizing electrodes occupy one or more vertical locations down one borehole while the potential electrodes occupy one or more vertical locations down one or more additional boreholes. Where a subsurface mine working or other cavity is available for disposition of current electrodes, with the potential electrodes at surface, a drift-to-surface method becomes available. When the current electrodes are at the surface and the potential electrodes are in the mine working or other cavity, then a surface-to-drift method becomes available. By analogy, boreholes or mine shafts afford drift-to-borehole or borehole-to-drift arrangements.
These known survey techniques offer improved lateral and vertical resolution and increased depth of exploration compared to surface techniques and to well-logging techniques. Nevertheless, each of the aforedescribed configurations of electrodes is limited in its ability to provide sufficient data to yield maximum vertical and horizontal resolution of subsurface bodies via use of equation (2).
When all four 1/r.sub.n terms in equation (2) are of comparable value, the electrodes are considered to be placed in the dipole-dipole array. When one current electrode is placed on the surface a great horizontal distance, "at infinity," from a borehole, two of the r.sub.n terms of equation (2) become very small, so that, by convention ##EQU2## Such an arrangement of electrodes will be referred to as the pole-dipole array. In the pole-pole array, both one current electrode and one potential electrode are placed on the surface a great horizontal distance away ("at infinity"). Equation (2) then reduces, by convention to: ##EQU3##
When the earth is inhomogeneous and/or anisotropic, both apparent resistivity and induced polarization can be calculated along surface, borehole, or subsurface profiles when the resistivity and/or induced polarization distributions in the subsurface are assumed. This mathematical procedure is referred to as forward modeling; i.e., a prediction of apparent resistivity and/or induced polarization along surface, borehole, or subsurface profiles or surfaces is made from an assumed model of the earth.
As an example, equation (4) pertains to the pole-pole array on the surface of a homogeneous half-space. When the electrodes are below the surface of the earth, images of each current electrode occur above the surface of the earth. Then equation (4) exhibits the form: ##EQU4## where r.sub.i is the distance from the potential pole to the image of the current pole.
Inverse solutions are known whereby a least-squares fit is performed between measured and computed values of .rho..sub.a versus pertinent electrode spacings. A Weighted least-squares sum may be written: ##EQU5## where (x.sub.i).rho..sub.a.sup.OBS is the observed or measured value of apparent resisitivity at location x.sub.i and (x.sub.i,.rho.) .rho..sub.a.sup.CAL is the calculated apparent resistivity at location x.sub.i due to the theoretical model defined by the parameters p. The n-dimensional parameter vector .rho. is comprised of such quantities as the resistivities of the overburden, host rock, and the several inhomogeneities inherent in the earth; it may also include depths, dips, and orientations of interfaces, depth extents and strikes of inhomogeneities, strike lengths, and thicknesses of inhomogeneities, and such other geometrical and physical property factors as may be inherent in the model selected to represent the real earth. The n-dimensional parameter vector p is varied through numerical automation until equation (6) is minimized. The quantity Var (x.sub.i).rho..sub.a.sup.OBS is the statistical variance of the observed data points.
Induced polarization parameters include: ##EQU6## where .rho..sub.LF is the apparent resistivity measured at some low frequency, e.g., 0.1 Hz, .rho..sub.HF is the apparent resistivity measured at some high frequency, e.g., 1Hz; and EQU .phi.=phase of measured voltage with respect to impressed current; (8)
both of which are measured in the frequency domain, and EQU M (chargeability)=MV.multidot.s/V, measured in the time domain, (9)
where MV.multidot.s is an area under a received voltage decay curve after the transmitting current is cut off, while V is the voltage observed at the receiver (potential electrodes) when the transmitting current (current electrodes) is on.